Japan “Sangaku” Research Institute

Problem In a quadrilateral $ABCD$, when opposite $∠A$ and $∠C$ are equal, the bisectors of another pair of opposite $∠B$ and $∠D$ are parallel to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $BM$ and $DN$…

Problem For a quadrilateral $ABCD$, if the bisectors of $∠A$ and $∠C$ intersect on the diagonal $BD$, then the bisectors of $∠B$ and $∠D$ intersect on the diagonal $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution If the…

Problem When the bisectors of the four angles of a quadrilateral pass through the same point, the sum of the lengths of one pair of opposite sides of the quadrilateral is equal to the sum of the lengths of the other pair of opposite sides. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…

Problem If the longest side of a quadrilateral $ABCD$ is $AD$ and the shortest side is $BC$, then $∠BCD$ is greater than $∠BAD$ and $∠ABC$ is greater than $∠ADC$: $$∠BCD>∠BAD \qquad and \qquad ∠ABC>∠ADC.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Let $ABCD$ be any quadrilateral, and let there be an interior point $O$ which is not the intersection of the diagonals; then the sum of $OA, \ OB, \ OC$ and $OD$ is greater than the sum of both diagonals $AC$ and $BD$: $$OA+OB+OC+OD＞AC+BD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…

Problem In the following $(1)$ to $(4)$, for the condition on the left about a quadrilateral, if the condition on the right is a necessary and sufficient condition, answer $A$; if it is a necessary but not sufficient condition, answer $B$; if it is a sufficient but not necessary condition, answer $C$; if it is…

Problem The perimeter of a quadrilateral is greater than the sum of its diagonals but less than twice the sum. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△ABC$ and $△ACD$, $$AC<AB+BC,$$ $$AC<CD+DA,$$ $$BD<BC+CD,$$ $$BD<AB+DA,$$ $$∴ \ 2(AC+BD)<2(AB+BC+CD+DA),$$…

Problem The sum of the interior angles of a quadrilateral is equal to four right angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution As shown in the diagram above, if you draw a diagonal $AC$ on a quadrilateral…